# The abundance of primordial black holes depends on the shape of the inflationary power spectrum

###### Abstract

In this letter, combining peak theory and the numerical analysis of gravitational collapse in the radiation dominated era, we show that the abundance of primordial blacks holes, generated by an enhancement in the inflationary power spectrum, is extremely dependent upon the shape of the peak. Given the amplitude of the power spectrum we show that the density of primordial black holes generated from a narrow peak, is exponentially smaller compared to the case of a broad peak. Specifically, for a top-hat profile of the power spectrum in Fourier space, we find that in order to consider primordial black holes as the whole of dark matter, one needs the amplitude of the power spectrum to be an order of magnitude smaller than what has been used so far. In the case of a narrow peak, one would instead need a much larger amplitude, which in many cases invalidates the perturbative analysis of cosmological perturbations. Finally, we show that, although the critical collapse gives a broad mass spectrum, the density of primordial black holes is essentially peaked at a single mass value.

## I Introduction

Combination of direct and indirect constraints seems to bound the amount of primordial black holes (PBHs) to account for at most a of the total abundance of dark matter (DM) carr ; seljak . However, indirect constraints on PBHs are due to complicated astrophysical dynamics and so are extremely model dependent. Therefore, further studies are required before taking these constraints as robust limitations of the PBHs abundance. For this reason it is interesting to consider these objects as a possible explanation of the total amount of DM carr : excluding indirect constraints, PBHs could be the whole of DM in the range . In any cases, the presence of PBHs, whether they consist the whole of the DM or not, is a generic feature of inflation and therefore their observation or absence could shed light on the early Universe.

The observational absence of isocurvature perturbations and non-Gaussianities in the latest cosmic microwave background data (the CMB spectrum) is in favour of single field models of inflation. In this context it has been proposed by garcia1 that a flattening of the inflationary potential, after the generation of the observed CMB curvature perturbations, might greatly enhance the power spectrum at scales smaller than the CMB (see also others ) so to generate a non-negligible abundance of PBHs.

While PBHs will always form by statistical fluctuations of curvature perturbations generated during inflation, their abundance is related to the specific dynamics of the inflaton, as seen recently in germani and Hu . Additionally, the abundance of PBHs depends upon the ratio between the amplitude of the inflationary power spectrum and a threshold . This threshold is related to the minimal amplitude of an initial curvature perturbation eventually collapsing into a black hole.

Recently, there has been some confusion about the correct estimate of : for example, in garcia1 and garcia2 , a rather small value of has been mistakenly equated to the analytical estimate of the critical value for the integrated density perturbations harada . A larger value of malik ; germani was obtained by translating incorrectly the critical amplitude of the integrated density perturbations into , which has been considered in ballesteros (in the realm of effective field theories) and in cicoli (within explicit string theory realisations).

In this paper, using peak theory peak , we show that all previous estimates of are actually inconsistent with the numerical simulations of PBH formation Niemeyer ; Shibata ; muscop ; harada2 ; muscoin , whether or not the PBHs consist the whole of DM. The key point is that the threshold is not universal but instead strongly depends upon the shape of the inflationary power spectrum.

In the following analysis, we provide the correct procedure to calculate the PBHs abundance from a given inflationary power spectrum.

## Ii Relations between gauges

We will start by considering a linear scalar perturbation on a FRW metric associated to an over-density . In the Kodama-Sasaki gauge (KS) ks the metric can be written as

and at super-horizon scales, where curvature perturbations are frozen and spatial gradients are small, this metric can be approximated as

(1) |

For a spherically symmetric perturbation, the above metric can be written as the following local FRW metric

(2) |

and the connection between the two metrics is given at linear order by

(3) |

where .

The line element (2), is the asymptotic metric used to set initial conditions for numerical simulations of a spherical over-density collapsing within a FRW universe muscop ; harada2 ; muscoin . Although here we are using the linear approximation, PBH formation requires, at initial time, a non-linear value of in the region where the black hole is going to form. Nevertheless, assuming Gaussian statistics for primordial perturbations, using peak theory peak , we will able to infer the non-linear, rare, initial conditions from the linear analysis.

A note of warning here is necessary: given the fact that PBH formations are rare events, there could be other ingredients playing an important role into the calculation of PBH abundance. For example the non-Gaussian contributions to the statistics of primordial curvature perturbations kehagias . However, it has been shown that non-Gaussianities might only give a small contribution for any smooth inflationary dynamics sasaki .

Finally, another gauge necessary to make the connection between the results of the numerical simulations and the analytical calculations of primordial perturbations, is the uniform density gauge () given by

(4) | |||||

(5) |

At super-horizon scales, this metric can be approximated as maldacena

(6) |

Therefore, in this regime, and, the KS and the uniform gauges, are related by

(7) |

## Iii The energy density profile

By using the Einstein equations, in the KS gauge at super-horizon scales, one finds that the energy-density perturbation, during the radiation dominated epoch, is expressed in terms of the curvature profile as

(8) |

where is the Hubble “constant”, the background energy-density and .

The translation of the above relation in is then easily obtained using equation (3),

(9) |

where . We now consider the Fourier form of (9)

(10) |

and using peak theory peak , we are able to reconstruct the mean profile of a rare over-dense peak in real space.
Since is assumed to be a Gaussian random variable with zero mean value, because of (10), also is approximately
a Gaussian random variable^{1}^{1}1In light of Jaume we will briefly discuss this approximation at the end of the paper..

The variance of is

(11) | |||||

(12) |

where we have used a standard definition of the curvature perturbation power spectrum , while is the power spectrum of the over-density. Finally, we can now define the momenta of as

The density of PBHs at the moment of formation must be much smaller than the density of the background radiation, otherwise they will dominate the present Universe when it becomes matter dominated. For this reason the peaks generating PBHs are rare and, to a good approximation, can be considered spherical. In this case, the two-point correlation function of in real space is

(13) |

The observed super-horizon density profile is constructed by using the multivariate Gaussian distribution of the (real space) random field . Following peak the super-horizon averaged density profile per given , implying peaks are rare, is

(14) |

where and is the amplitude of the over-density at the centre of the profile. Note that does not need to be “small”, therefore, once the dependence of (14) is constructed from linear theory, the same profile can also be used in the non-linear regime relevant for PBH formation.

In this limit the number density of peaks corresponding to a given amplitude , in the comoving volume, is

(15) |

where and, at super-horizon scales, is time independent. The critical value discriminates between perturbations collapsing into a black hole () and perturbations dispersing into the expanding Universe ().

As customary, we will assume that the number of high peaks generated at super-horizon scales is conserved also at sub-horizon scales. Then the number density of PBHs in physical space, at the moment of formation, is given by

where is the time when the PBHs are formed. Note that is independent upon the rescaling of the scale factor and thus the same is also valid for , as it should be. Finally, we are now able to define the density of PBHs of a given mass at formation to be

(16) |

The relative density of PBHs at formation with respect to the background energy-density is

(17) |

where and finally is the Planck mass. The lower limit corresponds to which is the mass of PBHs that would have been evaporated today.

Given , the PBH mass is well approximated by the scaling law of critical collapse Niemeyer ; muscop

(18) |

where for radiation , is a numerical coefficient that depends on the specific density profile and is the Horizon mass at the time , defined later on as the “horizon crossing time”.

Since exponentially decays with and , we extend the upper limit of (17) to infinity obtaining

(19) |

where

Numerical simulations show that is weakly dependent from muscoin , giving approximately , and therefore we take this factor out from .

Assuming that the horizon mass at formation is much larger than , otherwise no significant PBH abundance will be generated^{2}^{2}2We have in mind here that these PBHs will account for all, or for a significant part, of dark matter., one can approximate the previous integral with its saddle point:

(20) |

Note that, if the linear approximation applies () the density of PBHs is peaked at

(21) |

Finally, if we want to match the abundance of the PBHs with the observed DM, we would need , as can be seen for example in cicoli .

## Iv Calculation of the threshold

The amplitude of the perturbation , in the metric (2), can be related to the mass excess as follows

(22) |

where .

As explained in harada2 ; muscoin , a PBH is formed when the maximum of the compactness function is larger than a critical threshold, where is the areal radius. At super-horizon scales and it is time independent.

One can characterise the critical amplitude of the over-density to form a PBH by . Here, is the location of the maximum of , while is (with an abuse of language) the “horizon crossing” time, defined such that . Indeed, does not define the “true” horizon crossing of the perturbation as it is defined only by using super-horizon quantities. In any case, turns out to be a key parameter to compare different initial shapes of the initial over-densities in real space. Indeed, although the value of is shape dependent, the criteria to compute is shape independent, as shown in muscoin .

Given a particular curvature/density profile there is a critical value of (), and a corresponding critical (or a ), such that the collapse of the over-density is not bouncing back into the expanding Universe. Numerically, one finds that the range of is relatively small: . On the contrary, the relevant quantity to calculate the PBH abundance, , varies from to infinity, in a profile dependent way muscoin .

So far it has been used , calculated at the edge of the the over-density^{3}^{3}3Note also that in general ., in place of . This introduces a large error in the calculation of , as we shall see.

## V Results

We have so far discussed how to related the abundance of PBHs with the primordial power spectrum in the case of rare peaks, . We will see that generically . Thus, the approximation of rare peaks, implying spherical symmetry, is intimately related to the linearity of the mean primordial perturbations.

In the following, as benchmarks of power spectrums generated during inflation, we will consider the case of a narrow power spectrum, and the opposite case of a broad spectrum, simplified as a top-hat distribution.

### v.1 Narrow power spectrum

The first power spectrum we will consider is

(23) |

in the limit . In this case, one finds , and . Numerical simulations of PBH formation using a density profile obtained from (23), provide a critical muscoin and so .

To have an order of magnitude estimate, we can crudely approximate . For the whole of DM in PBHs of mass , we would need and therefore (it does not change significantly even up to ). Therefore, since to produce the seeds of PBHs from inflation one requires , there is only a little margin for this kind of spectrum to work. Finally, the PBHs formed by this spectrum are peaked at .

### v.2 Broad power spectrum

In this second case, we consider a top-hat spectrum with amplitude , extended in the range , with .^{4}^{4}4This range needs to be consistent with the gradient expansion for . Therefore, at the time of peak creation. In this case, , and . Numerical simulations using this power spectrum gives muscoin and so .

For we get , which is one order of magnitude smaller than the value previously quoted in the literature, e.g. germani ; ballesteros ; cicoli . Therefore, PBHs can be more likely produced than what has been reported in the literature so far. Finally, the PBHs formed by this spectrum, are peaked at a mass .

### v.3 Comparison to previous literature

Although the functional form of in eq. (20) differs from the one used in the literature, see for example carr2 , the largest error of previous analysis comes mainly from the discordant definitions of . There, the estimation of the PBHs abundance generated during the radiation era was incorrectly related to the value of . For a Mexican-hat profile one has muscop and that was taken as a universal critical value. Then, was related to at horizon crossing: by considering (10) at , it was used a critical . In figure 1 we plot the ratio of the abundances related to and in the approximation fixing as a function of . This figure clearly shows that the approach used previously is incorrect by many order of magnitudes. instead of

## Note added

At the full non-linear level, but still at super-horizon scales, depends non-linearly on and therefore, even if is a Gaussian variable, is not. The corrections to due to this fact are proportional to the higher correlations functions of . Since inflation is only consistent for , one would expect these non-gaussian corrections to be small, as we have assumed. Nevertheless, given a critical value , in a paper appeared the same day as ours Jaume , these corrections has been evaluated finding, compared to the linear case, an extra suppression factor for . It would be interesting to study the dependence of due to these non-linearities. This is however left for a future work.

###### Acknowledgements.

C.G. would like to thank Pierstefano Corasaniti and Licia Verde for discussions about peak theory and Vicente Atal for discussion on different inflationary models. IM would like to thank Misao Sasaki for discussions during the YITP long-term workshop “Gravity and Cosmology 2018”, YITP-T-17-02. The Authors would like to thank Jaume Garriga and Juan Garcia-Bellido for feedbacks on the previous version of this paper. CG is supported by the Ramon y Cajal program and by the national FPA2013-46570-C2-2-P and FPA2016-76005-C2-2-P grants. IM, and partially CG, are supported by the Unidad de Excelencia María de Maeztu Grant No. MDM-2014-0369.## References

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